Beginning with the familiar expression for frequency modulation (FM) synthesis, a system is examined/analyzed whereby the carrier frequency is modulated by ``looping back'' the output signal, with a feedback coefficient determining the amount of frequency deviation. ``Loopback FM'' is reminiscent of the previously coined ``Feedback FM'' technique, but with the feedback signal being applied to the carrier frequency (in the former) rather than the initial phase (as in the latter). Though both produce harmonic distortion, loopback FM has a less obvious phase modulation (PM) representation (the numerically advantageous implementation of FM) requiring integration of the instantaneous frequency, and also results in a shift of the sounding frequency which can be a potential source of frustration for some computer musicians. If the feedback coefficient is made time varying, the pitch will also change over time (glide) and further complications arise in the integration of frequency and PM representation. In this work the change of pitch is explored via an alternate representation of the system also having two variables, expressions for which are given in terms of corresponding FM/PM parameters. In addition to providing a mapping between desired pitch trajectories and FM parameters, this alternate representation allows for analytic (rather than numerical) integration of frequency, yielding a more precise expression for instantaneous phase in the PM representation. Also potentially musically useful is the ability to reverse the process by obtaining loopback FM parameters for specified pitch glide trajectories over time.
